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Due to nonzero intrinsic spin-orbit interaction in buckled honeycomb crystalstructures, silicene and germanene exhibit interesting topological properties,and are therefore candidates for the realization of the quantum spin Halleffect. We employ the Kane-Mele model to investigate the electron states inhexagonal silicene and germanene nanorings having either zigzag or armchairedges in the presence of a perpendicular magnetic field. We present results forthe energy spectra as function of magnetic field, the electron density of thespin-up and spin-down states in the ring plane, and the calculation of theprobability current density. The quantum spin Hall phase is found at the edgesbetween the nontrivial topological phase in silicene and germanene and vacuum.We demonstrate that the helical edge states in zigzag silicene and germanenenanorings can be qualitatively well understood by means of classical magneticmoments. However, this is not the case for comparable-sized armchair nanorings,where the eigenfunctions spread throughout the ring. Finally, we note that theenergy spectra of silicene and germanene nanorings are similar and that thedifferences between the two are mainly related to the difference in magnitudeof the spin-orbit coupling.

Helical edge states in silicene and germanene nanorings in perpendicular magnetic field